• Roberta Ghezzi (University of Burgundy, Dijon, France): Geometry and Analysis on Almost-Riemannian Manifolds.

Abstract: An almost-Riemannian structure on a n-manifold is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating n-tuple of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point, but in general it has rank < n on a nonempty set which is generically an embedded submanifold.

The first example is the Grushin space, which appeared in the context of hypoelliptic operators in the 70s. More recently, almost-Riemannian geometry arise as the natural framework to model problems of population transfer in quantum control systems.

We will investigate topological, metric and geometric aspects of almost- Riemannian manifolds and we will present recent results concerning diffusion processes. Time permitting, we will give the spectral analysis of the Laplace-Beltrami operator.

References:

A.A. Agrachev, U. Boscain, G. Charlot, R. Ghezzi, M. Sigalotti, Two-Dimensional Almost-Riemannian Structures with Tangency Points, Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire, 27(3) (2010) pp. 793-807.

U. Boscain, G. Charlot, M. Gaye, P. Mason, Local properties of almost-Riemannian structures in dimension 3, arXiv:1407.0610

U. Boscain, C. Laurent, The Laplace-Beltrami operator in almost-Riemannian geometry, Annales Institut Fourier (Grenoble) 63 (2013), no. 5, 1739-1770.

U. Boscain, D. Prandi, M. Seri, Spectral Analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds, arXiv: 1406.6578

• Tom Mestdag (Ghent University, Belgium): The inverse problem of the calculus of variations.

Abstract: The inverse problem we will discuss is the following one: given a system of second-order ordinary differential equations, under what circumstances can these equations be derived from a variational principle, i.e. when does there exist a regular Lagrangian function, such that its corresponding Euler-Lagrange equations are equivalent to the original equations (i.e. have the same solutions). In the introduction to his famous  paper `Solution of the inverse problem of the calculus of variations', published in 1941, Fields medalist Jesse Douglas said that the `problem indicated in the title is one of the most important hitherto unsolved problems of the calculus of variations'. Unlike his title suggests, Douglas gave the solution to this problem for two dimensions only, and the problem has not been solved (in the sense that Douglas solved it) for higher dimensions, either in general, or even in any particular subcase. This is not to say that no progress has been made in the intervening 70 years.

In the lectures, we will make use of a differential geometric approach to the problem, the calculus of derivations of forms along a map. We will focus on some integrability aspects, on some specific subcases, on an extension of the problem to systems with non-conservative forces, and on the related problem of metrizability in Finsler geometry.

Some general references to the inverse problem are:

[1] I. M. Anderson and G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations, Mem. Amer. Math. Soc. 98 (1992).

[2] J. Grifone and Z. Muzsnay, Variational Principles for Second-order Differential Equations, (World Scientific 2000).

[3] O. Krupkova and G. E. Prince, Second order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations, in: Handbook of Global Analysis, D. Krupka and D. Saunders eds. (Elsevier 2008) 837–904.

[4] R. M. Santilli, Foundations of Theoretical Mechanics I. The Inverse Problem in Newtonian Mechanics" (Spinger-Verlag 1978)

[5] W. Sarlet, G. Thompson and G.E. Prince, The inverse problem in the calculus of variations: the use of geometrical calculus in Douglas's analysis, Trans. Amer. Math. Soc. 354, 2897-2919 (2002)

• David Martínez Torres (PUC-Rio, Brazil): Geometry and topology of coadjoint orbits.

Abstract: Starting from Hamiltonian actions, we shall look at all coadjoint orbits (Poisson geometry) to obtain a geometric incarnation of basic facts from Lie theory of compact Lie groups. We shall also look at coadjoint orbits of compact groups individually (symplectic geometry), and discuss their topology and complex geometry. Time permiting, we shall recall connections with representation theory, and discuss features of coadjoint orbits of non-compact semisimple Lie groups.

List of short talks

• Alexis Arnaudon (Imperial College London). Lagrangian reduction for completely integrable systems.
• Marta Farré (Instituto de Ciencias Matemáticas). Inverse problem in discrete mechanics.
• Irina Mihaela Gheorghiu (University de Zaragoza). The virial theorem for nonholonomic systems.
• Hassan Jolany (University of Lille 1). Song-Tian theory in Birational Geometry and MMP.
• Jorge Alberto Jover Galtier (University of Zaragoza). Kähler-Lie systems and geometric quantum mechanics.
• Nicolás Martínez (IMPA). Higher-Poisson structures, reduction and field theory.
• Tamás Milkovszki (University of Debrecen). Invariant metrizability and projective metrizability on Lie groups.
• Mikolaj Rotkiewicz (Warsaw University). Prototypes of higher algebroids with applications to variational calculus.
• Nicola Sansonetto (Universiti di Padova). Complete integrability, the Hamilton-Jacobi equation and nonholonomic systems.
• Goedele Waeyert (Ghent University). Lifted tensors and Hamilton-Jacobi separability.

Book of abstracts here.